This deep dive covers Special Cases and Patterns within Quadratic Expressions for GCSE Mathematics. Revise Quadratic Expressions in Algebra for GCSE Mathematics with 12 exam-style questions and 22 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 6 of 8 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Special Cases and Patterns
Perfect Square Trinomials
These follow the pattern: x² + 2ax + a² = (x + a)²
Recognition tip: The constant term is the square of half the coefficient of x
Example: x² + 12x + 36
Half of 12 = 6, and 6² = 36 ✓
Therefore: x² + 12x + 36 = (x + 6)²
Difference of Two Squares
Pattern: x² - a² = (x + a)(x - a)
Recognition tip: Two square terms subtracted
Example: x² - 25 = x² - 5² = (x + 5)(x - 5)
When Factorising Isn't Possible
Not all quadratics can be factorised using integers
Example: x² + x + 1
Need two numbers that add to 1 and multiply to 1
No integer solutions exist - this quadratic is "irreducible" over integers
Practice questions for Quadratic Expressions
Expand (x + 5)².
Explain how to recognise whether x² + 12x + 36 is a perfect square trinomial, and write it in factorised form.